let n be Element of NAT ; :: thesis: for K being Field
for M1 being Matrix of n,K st M1 is anti-circular holds
- M1 is anti-circular
let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is anti-circular holds
- M1 is anti-circular
let M1 be Matrix of n,K; :: thesis: ( M1 is anti-circular implies - M1 is anti-circular )
A1: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
assume M1 is anti-circular ; :: thesis: - M1 is anti-circular
then consider p being FinSequence of K such that
A2: len p = width M1 and
A3: M1 is_anti-circular_about p ;
set r = - p;
p is Element of (len p) -tuples_on the carrier of K by FINSEQ_2:92;
then A4: - p is Element of (len p) -tuples_on the carrier of K by FINSEQ_2:113;
A5: Indices (- M1) = [:(Seg n),(Seg n):] by MATRIX_0:24;
A6: width M1 = n by MATRIX_0:24;
A7: for i, j being Nat st [i,j] in Indices (- M1) & i <= j holds
(- M1) * (i,j) = (- p) . (((j - i) mod (len (- p))) + 1)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (- M1) & i <= j implies (- M1) * (i,j) = (- p) . (((j - i) mod (len (- p))) + 1) )
assume that
A8: [i,j] in Indices (- M1) and
A9: i <= j ; :: thesis: (- M1) * (i,j) = (- p) . (((j - i) mod (len (- p))) + 1)
((j - i) mod n) + 1 in Seg n by A5, A8, Lm3;
then A10: ((j - i) mod (len p)) + 1 in dom p by A2, A6, FINSEQ_1:def 3;
(- M1) * (i,j) = - (M1 * (i,j)) by A1, A5, A8, MATRIX_3:def 2
.= (comp K) . (M1 * (i,j)) by VECTSP_1:def 13
.= (comp K) . (p . (((j - i) mod (len p)) + 1)) by A3, A1, A5, A8, A9
.= (- p) . (((j - i) mod (len p)) + 1) by A10, FUNCT_1:13 ;
hence (- M1) * (i,j) = (- p) . (((j - i) mod (len (- p))) + 1) by A4, CARD_1:def 7; :: thesis: verum

end;

A11: width (- M1) = n by MATRIX_0:24;
A12: len (- p) = len p by A4, CARD_1:def 7;
then A13: dom (- p) = Seg (len p) by FINSEQ_1:def 3;
for i, j being Nat st [i,j] in Indices (- M1) & i >= j holds
(- M1) * (i,j) = (- (- p)) . (((j - i) mod (len (- p))) + 1)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (- M1) & i >= j implies (- M1) * (i,j) = (- (- p)) . (((j - i) mod (len (- p))) + 1) )
assume that
A14: [i,j] in Indices (- M1) and
A15: i >= j ; :: thesis: (- M1) * (i,j) = (- (- p)) . (((j - i) mod (len (- p))) + 1)
A16: ((j - i) mod n) + 1 in Seg n by A5, A14, Lm3;
(- M1) * (i,j) = - (M1 * (i,j)) by A1, A5, A14, MATRIX_3:def 2
.= (comp K) . (M1 * (i,j)) by VECTSP_1:def 13
.= (comp K) . ((- p) . (((j - i) mod (len p)) + 1)) by A3, A1, A5, A14, A15
.= (- (- p)) . (((j - i) mod (len p)) + 1) by A2, A6, A13, A16, FUNCT_1:13 ;
hence (- M1) * (i,j) = (- (- p)) . (((j - i) mod (len (- p))) + 1) by A4, CARD_1:def 7; :: thesis: verum

end;

then - M1 is_anti-circular_about - p by A2, A6, A11, A12, A7;
then consider r being FinSequence of K such that
A17: ( len r = width (- M1) & - M1 is_anti-circular_about r ) ;
take r ; :: according to MATRIX16:def 10 :: thesis: ( len r = width (- M1) & - M1 is_anti-circular_about r )
thus ( len r = width (- M1) & - M1 is_anti-circular_about r ) by A17; :: thesis: verum